nLab differential algebroid

Idea

A differential algebroid is an oidification of the concept of differential algebra.

Definitions

Let $K$ be a commutative ring and $C$ be a $K$-linear category. Then $C$ is a $K$-differential algebroid if for every hom-$K$-module $Mor(a,b)$ there is a $K$-linear morphism $d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b)$ such that for all objects $a,b,c\in Ob(C)$, morphisms $f:Mor(a,b)$ and $g:Mor(b,c)$, and $K$-linear morphisms $d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b)$, $d_{Mor(b,c)}:Mor(b,c) \to Mor(b,c)$, and $d_{Mor(a,c)}:Mor(a,c) \to Mor(a,c)$, a generalised Leibniz rule is satisfied:

$$d_{Mor(a,c)}(g \circ f) = d_{Mor(b,c)}(g) \circ f + g \circ d_{Mor(a,b)}(f) \,.$$

If all three objects are the same, this reduces down to the Leibniz rule for a derivation.

Examples

• A differential algebra is a differential algebroid with only one object.

Related concepts

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory